Channel error rate optimization using Markov codes

ABSTRACT

In one embodiment, a system provides for optimizing an error rate of data through a communication channel. The system includes a data generator operable to generate a training sequence as a Markov code, and to propagate the training sequence through the communication channel. The system also includes a Soft Output Viterbi Algorithm (SOVA) detector operable to estimate data values of the training sequence after propagation through the communication channel. The system also includes an optimizer operable to compare the estimated data values to the generated training sequence, to determine an error rate based on the comparison, and to change the training sequence based on the Markov code to lower the error rate of the data through the communication channel.

CROSS REFERENCE TO RELATED APPLICATIONS

This patent application claims priority to, and thus the benefit of anearlier filing date from, U.S. Provisional Patent Application No.62/375,239 (filed Aug. 15, 2016), the entire contents of which arehereby incorporated by reference.

SUMMARY

In one embodiment, a system provides for optimizing an error rate ofdata through a communication channel. The system includes a datagenerator operable to generate a training sequence as a Markov code, andto propagate the training sequence through the communication channel.The system also includes an equalizer/Soft Output Viterbi Algorithm(SOVA) detector operable to estimate data values of the trainingsequence after propagation through the communication channel. The systemalso includes an optimizer operable to compare the estimated data valuesto the generated training sequence, to determine an error rate based onthe comparison, and to change the training sequence based on the Markovcode to lower the error rate of the data through the communicationchannel.

The various embodiments disclosed herein may be implemented in a varietyof ways as a matter of design choice. For example, some embodimentsherein are implemented in hardware whereas other embodiments may includeprocesses that are operable to implement and/or operate the hardware.Other exemplary embodiments, including software and firmware, aredescribed below.

BRIEF DESCRIPTION OF THE FIGURES

Some embodiments are now described, by way of example only, and withreference to the accompanying drawings. The same reference numberrepresents the same element or the same type of element on all drawings.

FIG. 1 is a block diagram of an exemplary system for optimizing an errorrate of a channel.

FIG. 2 is a flowchart illustrating an exemplary process operable withthe system of FIG. 1.

FIG. 3 is an exemplary Voronoi plot illustrating the decision regionsfor the optimization of an error rate for a memoryless channel.

FIG. 4 is a block diagram of an exemplary data flow of the system ofFIG. 1.

FIGS. 5-7 are graphs illustrating exemplary simulation results of thesystem of FIG. 1.

FIG. 8 is a block diagram of an exemplary computing system in which acomputer readable medium provides instructions for performing methodsherein.

DETAILED DESCRIPTION

The figures and the following description illustrate specific exemplaryembodiments. It will thus be appreciated that those skilled in the artwill be able to devise various arrangements that, although notexplicitly described or shown herein, embody certain principles and areincluded within the scope of this disclosure. Furthermore, any examplesdescribed herein are intended to aid in understanding the principles ofthe embodiments and are to be construed as being without limitation tosuch specifically recited examples and conditions. As a result, theembodiments are not limited to the examples described below.

The capacity of a communication channel was formalized by Claude Shannonin his 1948 treatise “A Mathematical Theory of Communication”. In thattreatise, Shannon provided a formula to compute channel capacity basedon the maximization of mutual information between a channel input andthe channel output over source probability distributions. The channelcapacity imposes a fundamental bound on the maximum throughputperformance of reliable communication with any error correcting code.

Computing the capacity of a communication channel is an interestingtheoretical problem in its own right. One example includes theBlahut-Arimoto algorithm which has been devised to compute channelcapacity of discrete memoryless channels as well as channels withmemory, such as the magnetic recording channel of a disk drive. However,knowing the theoretical channel capacity does not immediately render howto achieve a maximum throughput with a practical error correcting code,which is a primary focus of coding theory. More recently, codingtheorists have designed error correcting codes to closely approach theShannon limit of channel capacity for certain channels.

Based on the notion of channel capacity, systems and methods presentedherein are operable to optimize a channel input (i.e., a source)probability distribution function (PDF) to maximize the capacity of thechannel. In this regard, a new source optimization is implemented basedon a previously unused metric, such as the detection error rate(BER/SER). The problem employs a constraint on the minimum source rateH(X)≥h₀ so as to prevent a trivial solution. However, h₀ can be viewedas free design parameter to allow for more design flexibility. Anotheradvantage of using a metric such as the SER is that it uses a simplerdetector (Viterbi or SOVA detector) as opposed to a soft output symboldetector, such as a BCJR detector (Bahl-Cocke-Jelinek-Raviv detector),which is used for the capacity metric.

With this in mind, systems and methods presented herein provide for theoptimization of error rates in a channel, such as bit error rates (BERs)and symbol error rates (SERs). For example, solutions for channels, withor without memory, are provided herein along with the implementation ofan iterative algorithm to optimize the error rate for the channel.

FIG. 1 is a block diagram of an exemplary system 100 for optimizing anerror rate of a channel 103. As mentioned, communication channels suchas 103 have typically been evaluated based their capacities. That is,previous systems measured the maximum throughput of effectivecommunication of data through the channel. Then, error correction codescould be implemented based on the channel capacity to ensure that thedata transmitted through the channel could be recovered.

However, measuring the channel capacity does not always, in practice,accurately determine the amount of recoverable data that may passthrough the channel. And, the channel's characteristics may change overtime and/or even across its physical properties. For example, a harddisk drive (e.g., a channel with memory) may comprise a plurality ofsectors where data is stored. Each sector of the hard disk drive may bethought of as unique channel because, among other reasons, each sectormay comprise different physical properties from the next sector.Accordingly, each sector on the hard disk drive may vary in terms ofcapacity from one sector to the next.

The system 100 optimizes the data throughput of the channel 103 byoptimizing the error rate of the channel 103 with a training sequence.Once the error rate of data through the channel 103 is optimized, actualdata can be configured to mimic the data of the training sequence so asto reduce its error rate through the channel 103 and thereby effectivelyreduce the amount of error correction used to recover the data from thechannel 103.

The system 100 starts the optimization by generating an initial sourcemodel 101 that is fed into a source generator 102. The source generator102 generates the training sequence x_(n) based on the initial sourcemodel 101 and propagates the training sequence x_(n) through the channel103. The channel 103, having an effect on the training sequence x_(n)(i.e., the channel 103 introduces errors), results in a noisyobservation sequence y_(n) that is processed by an equalizer/SOVAdetector 104.

The equalizer/SOVA detector 104 is a variant of the classical Viterbialgorithm. SOVA detectors have been used for years in the disk driveindustry and can be implemented efficiently in a variety ways as amatter design choice. The equalizer/SOVA detector 104 differs from theclassical Viterbi algorithm in that it uses a modified path metric whichtakes into account the a priori probabilities of input symbols, andproduces a “soft output” indicating the reliability of the decision.And, the equalizer provides equalization of the signal prior to SOVAdetection.

The equalizer/SOVA detector 104 processes the noisy training sequencey_(n) to generate an estimated sequence {circumflex over (x)}_(n) . Theoptimizer 105 compares the sequence {circumflex over (x)}_(n) to theoriginal training sequence x_(n) and outputs a new Markov source modelQ_(ij) to the source generator 102, while also tracking the error rate(e.g., BER and/or SER) through the channel 103 during each iteration.The quantity Q_(ij), represents the probability of starting from thedetector trellis states i and going to state j and fully describes theMarkov source model. The source generator 102 then changes its outputtraining sequence x_(n) based on the new source model Q_(ij) from theoptimizer 105. When the lowest error rate has been obtained (e.g., afterthe last iteration), the channel 103 has been characterized andoptimized in terms of its error rate such that data and error correctioncan be configured accordingly for propagation through the channel 103and recovered therefrom.

More specifically, once the error rate of the channel 103 is optimized,subsequent data through the channel 103 can be similarly configured tohave a comparable error rate. Thus, error correction codes can betailored for that identified error rate instead of genericallyblanketing the data with error correction. This process may result in alower data rate than previous systems. However, as less error correctionmay be used, the actual data throughput across the channel 103 mayincrease, in some instances significantly.

FIG. 2 is a flowchart illustrating an exemplary process 200 operablewith the system 100 of FIG. 1. In this embodiment, the source generator102 generates a training sequence as a Markov code, in the processelement 201. For example, the source generator 102 may initially receivean input from the initial source model 101 that prompts the sourcegenerator 102 to output a training sequence. In this regard, the sourcegenerator 102, acting on this initial source model 101, produces arandom process that undergoes transitions from one state to another on astate space. The probability distribution of a next state depends on thecurrent state and not on the sequence of events that preceded it. Thisspecific kind of finite memory property produces a Markov chain ofstates, or equivalently the training sequence x_(n), that is thenpropagated through the communication channel 103, in the process element202.

In the process element 203, the equalizer/SOVA detector 104 estimatesdata values of the input training sequence after the propagation throughthe communication channel 103. The optimizer 105 then compares theestimated data values to the generated training sequence from the sourcegenerator 102, in the process element 204, to determine an error rate ofthe data propagating through the channel 103, in the process element205. That is, the optimizer 105 compares the original training sequencex_(n) to the estimated values of the training sequence {circumflex over(x)}_(n). The optimizer 105 then determines the error rate based on thecomparison. The optimizer 105 compares that error rate with the previousiteration to determine if it has settled to a minimum, in the processelement 206.

If the error rate is lower, the optimizer 105 outputs the model Q_(ij)that operates as a new input model to the source generator 102. In thisregard, the optimizer 105 directs the source generator 102 to change thetraining sequence x_(n) via the change in the Markov source model in anattempt to lower the error rate of the data through the communicationchannel 103, in the process element 208. And, the process 200 returns tothe process element 201 to thus generate the new training sequence forthe channel 103. If, however, the error rate has settled to a minimum(e.g., nearly the same as the previous error rate), in the processelement 206, the optimizer 105 may use the last iteration of thetraining sequence, in the process element 207, as a basis forcharacterizing the channel 103 in terms of its error rate. In thisregard, the optimizer 105 may also use this model of the Markov chain toconfigure subsequent actual data through the channel 103 as well aserror correction with that data.

The embodiments shown and described herein may be implemented formemoryless channels and/or channels with memory. In some embodiments,the system 100 may be implemented as firmware within a disk drive system(i.e., channel with memory) to characterize the disk drive, and/orvarious sectors and regions thereof, in terms of its bit error rate.Then, coding techniques can be applied based on that bit error rate,including precoding and error correction coding, that configure how thedata is to be written to and recovered from the disk drive. Otherexemplary embodiments are shown and described in greater detail below.

Source Optimization Based on SER Metric

In this embodiment, Markov code source optimization is based on an errorrate metric of the channel 103. More specifically, the system 100 isoperable to find a Markov source model that minimizes the detectedsymbol error rate P_(e) with a lower limit constraint on the source rate(e.g., entropy) H(X). For example, the system 100 is operable to:

$\begin{matrix}{{{\min\limits_{P{(X)}}\mspace{14mu}{P_{e}\mspace{14mu}{such}\mspace{14mu}{that}\mspace{14mu}{H(X)}}} \geq h_{o}},} & {{Eq}.\mspace{14mu} 1.}\end{matrix}$where h₀ is a minimum allowed source rate. A constraint on the sourcerate H(X), in this embodiment, is used because it would obtain a trivialsolution of P_(e)=0 when H(X)=0 without it. Accordingly, a solution issought in the space of Markov source models Q_(ij)=μ_(i)P_(ij), whereP_(ij) is the Markov transition probability from trellis state i tostate j, and μ_(i) is the steady state probability of trellis state i.In this regard, a first solution for memoryless channels is presented asit is generally an easier problem than the case of channels with memory.Memoryless Channel

Consider the source optimization problem for a discrete memorylesschannel having an input alphabet

and an output alphabet

. The expression for the detected SER is the expected value (e.g., over

) of the given that a particular symbol x from

was transmitted. In other words,P _(e) =

P(x)E(x),  Eq. 2.where E(x)=P({circumflex over (x)}≠x|x transmitted) and denotes theconditional SER given that the symbol x is transmitted and detected as{circumflex over (x)}≠x. One example includes the maximum a posteriori(MAP) detector that is optimal in terms of the SER metric. It producesan output {circumflex over (x)} that maximizes the probabilityP(x|y)∝P(x)P(y|x) over all x∈

as follows:

$\begin{matrix}{\hat{x} = {\arg{\max\limits_{x}{{P(x)}{{P( y \middle| x )}.}}}}} & {{Eq}.\mspace{14mu} 3.}\end{matrix}$

Based on the probability density function P(x|y), the output alphabet

is partitioned into decision regions

(x) (e.g., “Voronoi cells”) for each input symbol x as shown in FIG. 3.Detection is achieved by determining the cell in which the receivedsignal y lies, resulting in E(x)=P(y∉

(x)) where E(x) is computed either analytically (e.g., for relativelysimple channels) or numerically. Summarizing, the source optimizationproblem can be stated as:

$\begin{matrix}{{\min\limits_{P{(x)}}{\sum\limits_{x}{{P(x)}{E(x)}}}},} & {{Eq}.\mspace{14mu} 4.}\end{matrix}$such thatH(X)

−Σ_(x) P(x)log₂ P(x)≥h ₀.  Eq. 5.

Assuming that the “E-factors” E(x) are fixed quantities, an optimizingP(X) can be found by solving the following set of nonlinear equations:P(x)=Z ⁻¹2^(−E(x)/λ),  Eq. 6.Z=Σ _(x)2^(−E(x)/λ),  Eq. 7.andh ₀=−Σ_(x) P(x)log₂ P(x),  Eq. 8.where the auxiliary parameter λ>0 is the Lagrange multiplier associatedwith the entropy constraint.

However, E(x) does not generally remain constant since they dependimplicitly on P(X), and altering it will cause E(x) to change too.Accordingly, the system 100 employs an iterative algorithm where, ineach iteration, the E-factors are computed based on the current sourcemodel from the source generator 102. The source generator 102 is thenupdated treating the E-factors as constant. Such is summarized in thealgorithm below:

-   -   First, initialize source P(X) (e.g., an initial input 101 of        independent, identically distributed “IID” random variables).    -   Then, repeat the following steps until convergence        -   Estimate E-factors E(x) for current source P(X); and        -   Treating E(x) as fixed, update P(X) to minimize P_(e).            Channels with Memory

For the general case of Markov source optimization for channels withmemory and for a given Markov source model Q_(ij)=μ_(i)P_(ij), the SERcan be linearly approximated in a manner similar to Equation 3 as:P_(e)=Σ_(i,j)Q_(ij)E_(ij),  Eq. 9.for suitable coefficients E_(ij). The estimation of these coefficientsis implemented via training. Therefore, the source optimization problemof Equation 1 becomes

$\begin{matrix}{\min\limits_{Q_{ij}}{\sum\limits_{i,j,}{Q_{ij}E_{ij}}}} & {{Eq}.\mspace{14mu} 10}\end{matrix}$such thatH(X)

−Σ_(i,j) Q _(ij) log₂ P _(ij) ≥h ₀.  Eq. 11.

Again, assuming that the E-factors E_(ij) are fixed quantities, it canbe shown that the optimal Markov source model can be obtained. Forexample, let λ>0 denote the as yet unknown Lagrange multiplierassociated with the entropy constraint. Then, construct a “noisy”adjacency matrix A_(ij)=2^(−E) ^(ij) ^(/λ), and compute the optimalMarkov model as follows:

$\begin{matrix}{{\mu_{i} = \frac{u_{i}v_{i}}{\sum\limits_{i}{u_{i}v_{i}}}},{P_{ij} = {{\frac{v_{j}}{v_{i}}\frac{A_{ij}}{\lambda_{\max}}\mspace{14mu}{and}\mspace{14mu} C} = {\log_{2}\lambda_{\max}}}},} & {{Eq}.\mspace{14mu} 12.}\end{matrix}$where u and v are respectively the left and right eigenvectors of Acorresponding to its largest eigenvalue λ_(max). Afterwards, λ is chosento satisfy the source entropy constraint as follows:h ₀=−Σ_(i,j)μ_(i) P _(ij) log₂ P _(ij).  Eq. 13.This is a boundary solution in the sense that the source rate inequalityconstraint is in fact satisfied with an equality. The solution is foundusing a standard nonlinear equation solver.

Estimation of the E-factors E_(ij) can be obtained through training.Generally, in practice, the E-factors E_(ij) are difficult to computeanalytically. A stochastic estimate for E_(ij) using training data ispresented. In this regard, let the sequence x₁ ^(N) of length N denotethe input to the channel 103 and the sequence {circumflex over (x)}₁^(N) denote the output of the equalizer/SOVA detector 104 (i.e., thedetected training sequences). Also, let ε_(n)=Π(x_(n)≠{circumflex over(x)}_(n)) denote the error indicator sequence, and w_(H)(⋅) be theHamming weight function. Thus, as described in greater detail in thebelow:E _(ij) =E _(ij) ^(a) −E _(i) ^(b),  Eq. 14.where

$\begin{matrix}{E_{ij}^{a} = {\frac{1}{{NQ}_{ij}}{\sum\limits_{n}{{w_{H}( ɛ_{n - J}^{n + J - M} )}{{\mathbb{I}}( {S_{n - 1}^{n} = ( {i,j} )} )}}}}} & {{Eq}.\mspace{14mu} 15} \\{{E_{i}^{b} = {\frac{1}{N\;\mu_{i}}{\sum\limits_{n}{{w_{H}( ɛ_{n - J + 1}^{n + J - M} )}{{\mathbb{I}}( {s_{n} = i} )}}}}},} & {{Eq}.\mspace{14mu} 16.}\end{matrix}$for a window size parameter J that is typically chosen as a smallinteger.

In the summation of Eq. 15, all time indices n are observed where states_(n−)=i to the state s_(n)=j. For those indices, the number of errorsfound in the subsequence {ε_(n−J), . . . , ε_(n−J+M)} are accumulatedand a local neighborhood of time index n is identified. Likewise withEq. 16, the number of errors found in {ε_(n−J+1), . . . , ε_(n−J+M)} areaccumulated as the system 100 passes through state s_(n)=i at time n.Both expressions are relatively straightforward computations involvingpattern matching and accumulation of error weights. This algorithm canbe summarized as follows.

-   -   First, initialize Markov source Q_(ij) (e.g., an initial input        101 of IID random variables).    -   Then repeat the following steps until convergence:        -   Generate training sequence x₁ ^(N) from source Q_(ij);        -   Pass x₁ ^(N) through channel and detector to get {circumflex            over (x)}₁ ^(N);        -   Estimate E-factors E_(ij)=E_(ij) ^(a)−E_(i) ^(b) using Eqs.            15 and 16; and        -   Solve optimization problem of Eq. 10 to obtain a new Q_(ij).

FIG. 4 is a block diagram exemplarily summarizing functional aspects ofthe system 100 of FIG. 1. The source generator 102 generates thetraining sequence x₁ ^(N) based on the Markov model probabilitiesQ_(ij)=μ_(i)P_(ij). The source generator 102 can be implemented with aspecially designed Markov source encoder, such as an arithmetic encoder,or an idealized finite state machine which traverses the states of thetrellis based on the given transition probabilities P_(ij), formingconstrained bits 126. A pseudo random number generator can be used todrive such source generators.

Postcoding (e.g., postcoder 120) results in bits that are notconstrained. These bits are then transferred to a Low Density ParityCheck (LDPC) encoder 121 to add parity 129 for error correction. Theprecoder 122 provides returns the bits to their constrained form withparity bits 129. Precoding by the precoder 122 can be used to improveperformance. The BER in the postcoded domain 127 (e.g., using bits a_(n)and â_(n)) is minimized while maintaining the Markov property in theprecoded domain bits x_(n) (e.g., on the media), excluding the paritybits 129 added by the LDPC encoder 121.

The equalizer/SOVA detector 104 and the channel 103 generally constitutean environment from the point where the bits are transmitted or writtenonto media, such as a storage device (e.g., disk drive or solid statedrive—“SSD”), to the point where they are received and detected. TheLDPC decoder 123 and the precoder 124 return the sequence to its formafter the effects of the LDPC encoder 121 and the postcoder 120. Theonly change to the algorithm is that the system 100 computes the errorvector based on the postcoded bits, i.e., ε_(n)=Π(a_(n)≠â_(n)), againexcluding the parity bits 129. The Markov source decoder 125 thenextracts the Markov code to reveal underlying data.

The optimizer 105 computes the E-factors E_(ij) from the error vectorε_(n) and computes new source probabilities Q_(ij) by solving Equation10. This generally involves finding the largest eigenvalue of a matrixand solving nonlinear equations. Hence, the calculations may be moresuitable for implementation in firmware or as an offline calibration,but specialized hardware implementation is also possible. The freeparameter h₀ controls the source rate and yields an extra degree ofdesign flexibility.

The source generator 105 can be initialized with an IID model (e.g.,with all states being equally probable). Alternatively, in a magneticrecording system, different disc drives or even different zones on agiven disc may have different optimal Markov sources, owing to theirdifferent signal/noise statistics. These different Markov models may,however, be similar enough that the source generator 102 could startwith a relatively good/common initial solution such that the optimizer105 can “fine tune” the initial code separately for each drive or zonewithin a relatively small number of iterations.

The system 100 can be used with any alternative cost function (insteadof the SER) when it is expressed in a linear form similar to Equation 5.For example, the system 100 can minimize the conditional entropy H(X|Y)of input X given the output Y. This cost function represents theuncertainty in X given Y. To implement such, the system 100 setsE_(ij)=−T_(ij), where T_(ij) are the T-factors of a generalizedBlahut-Arimoto algorithm defined as:T _(ij)=Σ_(y) P(y|s _(n−1) ^(n)=(i,j))log₂ P(s _(n−1)^(n)=(i,j)|y)−Σ_(y) P(y|s _(n) =i)log₂ P(s _(n) =i|y)  Eq. 17.This approach maximizes the channel capacity but with the source ratebeing specified explicitly as: H(X)=h₀.

The following is an example Markov source designed for magneticrecording channels, specifically for a drive having a channel bitdensity (CBD) of 2.8. Here, the goal was to optimize the Markov codehaving the same source rate as a maximum transition run (MTR) constraintcode of K=3, but with a better BER performance. The MTR code, which iscommonly used in magnetic recording, limits the number of consecutivetransition in the source sequence to K=3 and has a source rate ofH_(MTR3)=0.94678 bits. The motivation for such a design is that it wouldbe a drop-in replacement for the MTR-3 code, due to its identical sourcerate, without requiring changes to other system parameters such as theCBD and LDPC code rates.

The BER minimizing source optimizer was implemented with the setting ofh₀=H_(MTR3)=0.94678 starting from an IID source model. FIG. 5 shows theBER vs. the signal to noise ratio (SNR) for the initial IID source(labeled as “iteration 0”) and after each iteration thereafter. There isa relatively large jump from iteration “0” to iteration “1” because thesource entropy H(X) changes from “1” (for the IID source) to h₀. In thesubsequent iterations, H(X)=h₀ (i.e., it remains constant) and the BERdrops only slightly with near convergence after two iterations.

FIG. 6 compares the BER for the Markov source (after convergence of thealgorithm) and the BER for the MTR-3 source. FIG. 7 shows a similarcomparison of the information rates for the new Markov source and theMTR-3 source. Both of the plots of FIGS. 6 and 7 show that the Markovsource offers a considerable gain over MTR-3, especially at lower SNRvalues.

Expression for P_(e)

One goal is to show that the probability of detection error P_(e) can beexpressed approximately as Eq. 9 to derive explicit formulas forEt_(ij). For example, let x₁ ^(N) and {circumflex over (x)}₁ ^(N) denotethe channel input and SOVA detected output training sequences. It canthen be assumed that the probability of x_(n) being detected incorrectlydepends strongly only on a “local neighborhood” consisting of a (2J+1)symbol subsequence Π_(n)=x_(n−J) ^(n+J) for a suitable window sizeparameter J as follows:P({circumflex over (x)} ₁ ^(N) ≠x _(n) |x ₁ ^(N))≈P({circumflex over(x)} ₁ ^(N) ≠x _(n)|Π_(n))

E(Π_(n)).  Eq. 18.

This approximation is used to limit the computational complexity of thealgorithm. By ergodicity, a statistical estimate for E(Π) for eachsymbol pattern Π of length 2J+1 is:

$\begin{matrix}{{{E(\Pi)} = {\frac{1}{{NP}(\Pi)}{\sum\limits_{{n:\Pi_{n}} = \Pi}{{\mathbb{I}}( {{\hat{x}}_{n} \neq x_{n}} )}}}},} & {{Eq}.\mspace{14mu} 19.}\end{matrix}$where Π(⋅) is the indicator function. Then, the SER expression is:P _(e)=Σ_(Π) P(Π)E(Π).  Eq. 20.

Note that the symbol subsequence Π=π₀ ^(2J) is equivalent to asubsequence α₀ ^(2J−M+1) of an M-bit detector trellis states α_(k)

π_(k) ^(k+M−1). Therefore, the Markov source model Q_(ij)=μ_(i)P_(ij)for the source generator 102 implies that:P(Π)=P(α₀ ^(2J−M+1))=Π_(k+0) ^(2J−M) Q _(α) _(k) _(,α) _(k+1) Π_(k=1)^(2J−M)μ_(α) _(k) ⁻¹,  Eq. 21.

Since the probability P(Π), and hence P_(e), are both nonlinearfunctions of the Markov probabilities, an approximate expression forP_(e) can be derived by linearizing near a given sourceQ_(ij)=μ_(i)P_(ij). In this regard, the derivative of log P(Π) at thecurrent model Q_(ij)=μ_(i)P_(ij) can be computed as:δ log P(Π)≈Σ_(k=0) ^(2J−M)δ log Q _(α) _(k) _(,α) _(k+1) −Σ_(k=1)^(2J−M)δ log μ_(α) _(k) .  Eq. 22.Therefore,

$\begin{matrix}{{\delta\;{P(\Pi)}} = {{P(\Pi)}{( {{\sum\limits_{k = 0}^{{2\; J} - M}\;\frac{\delta\; Q_{\alpha_{k},\alpha_{k + 1}}}{Q_{\alpha_{k},\alpha_{k + 1}}}} - {\sum\limits_{k = 1}^{{2\; J} - M}\;\frac{{\delta\mu}_{\alpha_{k}}}{\mu_{\alpha_{k}}}}} ).}}} & {{Eq}.\mspace{14mu} 23.}\end{matrix}$So, P_(e) can be approximated at the current model Q_(ij)=μ_(i)P_(ij) asfollows:

$\begin{matrix}{{\delta\; P_{e}} = {{\sum\limits^{\Pi}{{E(\Pi)}\delta\;{P(\Pi)}}} = {{\sum\limits_{\Pi}{{E(\Pi)}{P(\Pi)}( {{\sum\limits_{k = 0}^{{2\; J} - M}\;\frac{\delta\; Q_{\alpha_{k},\alpha_{k + 1}}}{Q_{\alpha_{k},\alpha_{k + 1}}}} - {\sum\limits_{k = 1}^{{2\; J} - M}\;\frac{{\delta\mu}_{\alpha_{k}}}{\mu_{\alpha_{k}}}}} )}} \equiv {{\sum\limits_{ij}{E_{ij}^{a}\delta\; Q_{ij}}} - {\sum\limits_{i}{E_{i}^{b}{{\delta\mu}_{i}.}}}}}}} & {{Eq}.\mspace{14mu} 24.}\end{matrix}$This yields the desired form for P_(e) as follows:P _(e)≈Σ_(ij) E _(ij) ^(a) Q _(ij)−Σ_(i) E _(i) ^(b)μ_(i)≡Σ_(ij) E _(ij)Q _(ij),  Eq. 25.where E_(ij)=E_(ij) ^(a)−E_(i) ^(b). With U(Π)

E(Π)P(Π), it can be readily seen that

$\begin{matrix}{{E_{ij}^{a} = {{\sum\limits_{\Pi}{{U(\Pi)}{\sum\limits_{k = 0}^{{2\; J} - M}\;\frac{{\mathbb{I}}( {{\alpha_{k} = i},{\alpha_{k + 1} = j}} )}{Q_{\alpha_{k},\alpha_{k + 1}}}}}} = {\sum\limits_{k = 0}^{{2\; J} - M}\;{\sum\limits_{{\Pi:\alpha_{k}^{k + 1}} = {({i,j})}}\frac{U(\Pi)}{Q_{ij}}}}}},\mspace{20mu}{and}} & {{Eq}.\mspace{14mu} 26.} \\{\mspace{79mu}{E_{i}^{b} = {\sum\limits_{k = 1}^{{2\; J} - M}\;{\sum\limits_{{\Pi:\alpha_{k}} = i}{\frac{U(\Pi)}{\mu_{i}}.}}}}} & {{Eq}.\mspace{14mu} 27.}\end{matrix}$

To illustrate, consider a binary-input channel with a trellis memory ofM=4 and a window size parameter of J=3. Then, Π is a 7-bit patternconsisting of 4 adjacent trellis states, Π≡(α, β, γ, δ). This means thatthe following expression for P(Π) is:

$\begin{matrix}{{P(\Pi)} = {{P({\alpha\beta\gamma\delta})} = {{\mu_{\alpha}P_{\alpha\beta}P_{\beta\gamma}P_{\gamma\delta}} \equiv {\frac{Q_{\alpha\beta}Q_{\beta\gamma}Q_{\gamma\delta}}{\mu_{\beta}\mu_{\gamma}}.}}}} & {{Eq}.\mspace{14mu} 28.}\end{matrix}$Assuming that E(Π)=E(αβγδ) has been estimated, Equations 26 and 27 maybe computed as Equations 29 and 30 as follows:

$\begin{matrix}\begin{matrix}{E_{ij}^{a} = {\frac{1}{Q_{ij}}( {{\sum\limits_{\gamma\delta}{U( {{ij}\;{\gamma\delta}} )}} + {\sum\limits_{\alpha\delta}{U( {\alpha\;{ij}\;\delta} )}} + {\sum\limits_{\alpha\beta}{U( {{\alpha\beta}\;{ij}} )}}} )}} \\{{\equiv {\frac{1}{Q_{ij}}( {\sum\limits_{\alpha\beta}\lbrack {{U( {{ij}\;{\alpha\beta}} )} + {U( {\alpha\;{ij}\;\beta} )} + {U( {{\alpha\beta}\;{ij}} )}} \rbrack} )}},{and}}\end{matrix} & {{Eq}.\mspace{14mu} 29.} \\\begin{matrix}{E_{i}^{b} = {\frac{1}{\mu_{i}}( {{\sum\limits_{\alpha\gamma\delta}{U( {\alpha\; i\;{\gamma\delta}} )}} + {\sum\limits_{\alpha\beta\delta}{U( {{\alpha\beta}\; i\;\delta} )}}} )}} \\{{\equiv {\frac{1}{\mu_{i}}( {\sum\limits_{\alpha\beta\gamma}\lbrack {{U( {\alpha\; i\;{\beta\gamma}} )} + {U( {\alpha\;\beta\; i\;\gamma} )}} \rbrack} )}},}\end{matrix} & {{Eq}.\mspace{14mu} 30.}\end{matrix}$where U(αβγδ)=E(αβγδ)P(αβγδ).Training Based Estimation of E_(ij)

In this section, it is illustrated how E(Π) and E_(ij) can be estimatedfrom training data. For example, let ε_(n)=Π(x_(n)≠{circumflex over(x)}₁ ^(N)) denote the bit error sequence and let w_(H)(⋅) denote theHamming weight. A stochastic (e.g., training based) estimate of E(Π) isthen given by

$\begin{matrix}{{E(\Pi)} = {\frac{1}{{NP}(\Pi)}{\sum\limits_{{n:\Pi_{n}} = \Pi}{{{\mathbb{I}}( {{\hat{x}}_{1}^{N} \neq x_{n}} )}.}}}} & {{Eq}.\mspace{14mu} 31.}\end{matrix}$Inserting Equation 31 into Equation 26 yields

$\begin{matrix}{{E_{ij}^{a} = {{\sum\limits_{k = 0}^{{2\; J} - M}\;{\sum\limits_{{\Pi:\alpha_{k}^{k + 1}} = {({i,j})}}\frac{{E(\Pi)}{P(\Pi)}}{Q_{ij}}}} = {{\frac{1}{{NQ}_{ij}}{\sum\limits_{k = 0}^{{2\; J} - M}\;{\sum\limits_{\Pi}{\sum\limits_{{n:\Pi_{n}} = \Pi}{{\mathbb{I}}( {{{\hat{x}}_{n} \neq x_{n}},{\alpha_{k}^{k + 1} = ( {i,j} )}} )}}}}} = {{\frac{1}{{NQ}_{ij}}{\sum\limits_{k = 0}^{{2\; J} - M}\;{\sum\limits_{n}{{\mathbb{I}}( {{{\hat{x}}_{n} \neq x_{n}},{s_{k + n - J + M - 1}^{k + n - J + M} = ( {i,j} )}} )}}}} = {\frac{1}{{NQ}_{ij}}{\sum\limits_{k = 0}^{{2\; J} - M}\;{\sum\limits_{n}{{{\mathbb{I}}( ɛ_{n + J - M - k} )}{{\mathbb{I}}( {s_{n - 1}^{n} = ( {i,j} )} )}}}}}}}}},} & {{Eq}.\mspace{14mu} 32.}\end{matrix}$which can be rewritten as

$\begin{matrix}{E_{ij}^{a} = {\frac{1}{{NQ}_{ij}}{\sum\limits_{n}{{w_{H}( ɛ_{n - J}^{n + J - M} )}{{{\mathbb{I}}( {s_{n - 1}^{n} = ( {i,j} )} )}.}}}}} & {{Eq}.\mspace{14mu} 33.}\end{matrix}$Similarly, inserting Equation 31 into Equation 27 and proceeding asbefore:

$\begin{matrix}{E_{i}^{b} = {\frac{1}{N\;\mu_{i}}{\sum\limits_{n}{{w_{H}( ɛ_{n - J + 1}^{n + J - M} )}{{{\mathbb{I}}( {s_{n} = i} )}.}}}}} & {{Eq}.\mspace{14mu} 34.}\end{matrix}$And then, the E-factors can be estimated using E_(ij)=E_(ij) ^(a)−E_(i)^(b).

The embodiments can take the form of an entirely hardware embodiment, anentirely software embodiment or an embodiment containing both hardwareand software elements. One embodiment disclosed herein is implemented insoftware, which includes but is not limited to firmware, residentsoftware, microcode, etc. FIG. 8 illustrates a computing system 300 inwhich a computer readable medium 306 may provide instructions forperforming any of the methods disclosed herein.

Furthermore, the embodiments can take the form of a computer programproduct accessible from the computer readable medium 306 providingprogram code for use by or in connection with a computer or anyinstruction execution system. For the purposes of this description, thecomputer readable medium 306 can be any apparatus that can tangiblystore the program for use by or in connection with the instructionexecution system, apparatus, or device, including the computer system300.

The medium 306 can be any tangible electronic, magnetic, optical,electromagnetic, infrared, or semiconductor system (or apparatus ordevice). Examples of a computer readable medium 306 include asemiconductor or solid state memory, magnetic tape, a removable computerdiskette, a random access memory (RAM), a read-only memory (ROM), arigid magnetic disk and an optical disk. Some examples of optical disksinclude compact disk-read only memory (CD-ROM), compact disk-read/write(CD-R/W) and DVD.

The computing system 300, suitable for storing and/or executing programcode, can include one or more processors 302 coupled directly orindirectly to memory 308 through a system bus 310. The memory 308 caninclude local memory employed during actual execution of the programcode, bulk storage, and cache memories which provide temporary storageof at least some program code in order to reduce the number of timescode is retrieved from bulk storage during execution. Input/output orI/O devices 304 (including but not limited to keyboards, displays,pointing devices, etc.) can be coupled to the system either directly orthrough intervening I/O controllers. Network adapters may also becoupled to the system to enable the computing system 300 to becomecoupled to other data processing systems, such as through host systemsinterfaces 312, or remote printers or storage devices throughintervening private or public networks. Modems, cable modem and Ethernetcards are just a few of the currently available types of networkadapters.

What is claimed is:
 1. A system for optimizing an error rate of datathrough a communication channel, the system comprising: a data generatoroperable to generate a training sequence as a Markov code, and topropagate the training sequence through the communication channel; aSoft Output Viterbi Algorithm (SOVA) detector operable to estimate datavalues of the training sequence after propagation through thecommunication channel; and an optimizer operable to compare theestimated data values to the generated training sequence, to determinean error rate based on the comparison, and to change the trainingsequence based on the Markov code to lower the error rate of the datathrough the communication channel.
 2. The system of claim 1, wherein:the channel is a storage media.
 3. The system of claim 1, wherein: thechannel is memoryless.
 4. The system of claim 1, wherein: the channel isa channel with memory comprising intersymbol interference (ISI).
 5. Thesystem of claim 1, wherein: the optimizer is further operable toconfigure subsequent data through the channel based on the determinederror rate.
 6. The system of claim 5, further comprising: an encoderoperable to configure the subsequent data with error correction coding.7. The system of claim 6, wherein: the encoder is further operable toconfigure the error correction coding based on the determined errorrate.
 8. A method of optimizing an error rate of data through acommunication channel, the method comprising: generating a trainingsequence as a Markov code; propagating the training sequence through thecommunication channel; estimating, with a Soft Output Viterbi Algorithm(SOVA) detector, data values of the training sequence after propagationthrough the communication channel; comparing the estimated data valuesto the generated training sequence; determining an error rate based onthe comparison; and changing the training sequence based on the Markovcode to lower the error rate of the data through the communicationchannel.
 9. The method of claim 8, wherein: the channel is a storagemedia.
 10. The method of claim 8, wherein: the channel is memoryless.11. The system of claim 8, wherein: the channel is a channel with memorycomprising intersymbol interference (ISI).
 12. The method of claim 8,further comprising: configuring subsequent data through the channelbased on the determined error rate.
 13. The method of claim 12, furthercomprising: configuring the subsequent data with error correctioncoding.
 14. The method of claim 13, further comprising: configuring theerror correction coding based on the determined error rate.
 15. Anon-transitory computer readable medium comprising instructions that,when executed by a processor, are operable to direct the processor tooptimize an error rate of data through a communication channel, theinstructions further directing the processor to: generate a trainingsequence as a Markov code source; propagate the training sequencethrough the communication channel; estimate, with a Soft Output ViterbiAlgorithm (SOVA) detector, data values of the training sequence afterpropagation through the communication channel; compare the estimateddata values to the generated training sequence; determine an error ratebased on the comparison; and change the training sequence based on theMarkov code to lower the error rate of the data through thecommunication channel.
 16. The computer readable medium of claim 15,wherein: the channel is a storage media.
 17. The computer readablemedium of claim 15, wherein: the channel is memoryless.
 18. The computerreadable medium of claim 15, wherein: the channel is a channel withmemory comprising intersymbol interference (ISI).
 19. The computerreadable medium of claim 15, further comprising instructions that directthe processor to: configure subsequent data through the channel based onthe determined error rate.
 20. The computer readable medium of claim 19,further comprising instructions that direct the processor to: configurethe subsequent data with error correction coding; and configure theerror correction coding based on the determined error rate.